# Chebyshev’sEquation ConsiderChebyshev’s equation ( 1 − x 2 ) y ″ − x y ′ + k 2 y = 0 Polynomial solutions of this differential equation are called Chebyshev polynomials and are denoted by T k (x). They satisfy the recursion equation T n + 1 ( x ) = 2 x T n ( x ) − T n − 1 ( x ) . Given that T 0 ( x ) = 1 and T 1 ( x ) = x , determine the Chebyshevpolynomials T 2 ( x ) , T 3 ( x ) and T 4 ( x ) . Verify that T 0 ( x ) , T 1 ( x ) , T 2 ( x ) , T 3 ( x ) , and T 4 ( x ) are solutions of the given differential equation. Verify the following Chebyshev polynomials. T 5 ( x ) = 16 x 5 − 20 x 3 + 5 x T 6 ( x ) = 32 x 6 − 48 x 4 + 18 x 2 − 1 T 7 ( x ) = 64 x 7 − 112 x 5 + 56 x 3 − 7 x ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378 ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378

#### Solutions

Chapter 16, Problem 16PS
Textbook Problem

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